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Raised-cosine filter
The raised-cosine filter is a particular electronic filter, frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference (ISI). Its name stems from the fact that the non-zero portion of the frequency spectrum of its simplest form ( \beta = 1 ) is a cosine function, 'raised' up to sit above the f (horizontal) axis. Mathematical description The raised-cosine filter is an implementation of a low-pass Nyquist filter, i.e., one that has the property of vestigial symmetry. This means that its spectrum exhibits odd symmetry about \frac{1}{2T} , where T is the symbol-period of the communications system. Its frequency-domain description is a piecewise function, given by: : H(f) = \begin{cases} T, & |f| \leq \frac{1 - \beta}{2T} \\ \frac{T}{2}\left+ \cos\left(\frac{\pi T}{\beta}\left[|f| - \frac{1 - \beta}{2T}\right\right)\right], & \frac{1 - \beta}{2T} < |f| \leq \frac{1 + \beta}{2T} \\ 0, & \mbox{otherwise} \end{cases} : 0 \leq \beta \leq 1 and characterised by two values; \beta , the roll-off factor, and T , the reciprocal of the symbol-rate. The impulse response of such a filter is given by: : h(t) = \mathrm{sinc}\left(\frac{t}{T}\right)\frac{\cos\left(\frac{\pi\beta t}{T}\right)}{1 - \frac{4\beta^2 t^2}{T^2}} , in terms of the normalized sinc function. Roll-off factor The roll-off factor, \beta , is a measure of the excess bandwidth of the filter, i.e. the bandwidth occupied beyond the Nyquist bandwidth of \frac{1}{2T} . If we denote the excess bandwidth as \Delta f , then: : \beta = \frac{\Delta f}{\left(\frac{1}{2T}\right)} = \frac{\Delta f}{R_S/2} = 2T\Delta f where R_S = \frac{1}{T} is the symbol-rate. The graph shows the amplitude response as \beta is varied between 0 and 1, and the corresponding effect on the impulse response. As can be seen, the time-domain ripple level increases as \beta decreases. This shows that the excess bandwidth of the filter can be reduced, but only at the expense of an elongated impulse response. \beta = 0 As \beta approaches 0, the roll-off zone becomes infinitesimally narrow, hence: : \lim_{\beta \rightarrow 0}H(f) = \mathrm{rect}(fT) where \mathrm{rect}(.) is the rectangular function, so the impulse response approaches \mathrm{sinc}\left(\frac{t}{T}\right) . Hence, it converges to an ideal or brick-wall filter in this case. \beta = 1 When \beta = 1 , the non-zero portion of the spectrum is a pure raised cosine, leading to the simplification: : H(f)|_{\beta=1} = \left \{ \begin{matrix} \frac{1}{2}\left+ \cos\left(\pi fT\right)\right, & |f| \leq \frac{1}{T} \\ 0, & \mbox{otherwise} \end{matrix} \right. Bandwidth The bandwidth of a raised cosine filter is most commonly defined as the width of the non-zero portion of its spectrum, i.e.: : BW = \frac{1}{2}R_S(1+\beta) Application When used to filter a symbol stream, a Nyquist filter has the property of eliminating ISI, as its impulse response is zero at all nT (where n is an integer), except n = 0 . Therefore, if the transmitted waveform is correctly sampled at the receiver, the original symbol values can be recovered completely. However, in most practical communications systems, a matched filter must be used in the receiver, due to the effects of white noise. This enforces the following constraint: : H_R(f) = H_T^*(f) i.e.: : |H_R(f)| = |H_T(f)| = \sqrt To satisfy this constraint while still providing zero ISI, a root-raised-cosine filter is typically used at each end of the communication system. In this way, the total response of the system is raised-cosine. References * Glover, I.; Grant, P. (2004). Digital Communications (2nd ed.). Pearson Education Ltd. ISBN 0-13-089399-4. * Proakis, J. (1995). Digital Communications (3rd ed.). McGraw-Hill Inc. ISBN 0-07-113814-5. External links *- Technical article entitled 'The care and feeding of digital, pulse-shaping filters' originally published in RF Design. Category:Linear filters Category:Telecommunication theory de:Raised-Cosine-Filter es:Flitro de coseno alzado it:Filtro a coseno rialzato ru:Приподнятый косинус (фильтр)